Please use this identifier to cite or link to this item: http://hdl.handle.net/11455/90451
標題: Tailored finite point methods and radial basis function collocation methods for solving partial differential equations and their applications
量身定做的有限點法與徑向基底函數配點法求解偏微分方程及其應用
作者: 蔡至清
Chih-Ching Tsai
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摘要: In this dissertation, we study the tailored finite point methods and the radial basis function collocation methods for solving partial differential equations including Schrdinger equation and the bioheat equation.o In the first part, we regard a time-independent Schrdinger equation as a singu-o larly perturbed problem. Under the assumption that the potential is nonnegative and homogeneous with degree greater than zero, we present two theorems for de- scribing the behavior of this problem on an unbounded domain and two conjectures on a bounded domain as the singular parameter approaches zero. We also propose two tailored finite point schemes to obtain the numerical solutions, which agree with the theoretical analysis for the two special cases. Moreover, we exploit an inexact method for the second tailored finite point scheme and compare the performance for various preconditioners with iterative algorithms. In the second part, we study the structure of vortex in a rotating Bose-Einstein condensate trapped by various external potentials by using the radial basis func- tion collocation method (RBFCM) to solve the coupled nonlinear Schrdinger equa-o tions (CNLSE) that describe Bose-Einstein condensate. It is well-known that the accuracy of the radial basis function collocation method is closely related to the shape parameter. At first, we give a rule to select the shape parameter when solving linear Schrdinger equation. Next, we numerically simulate the vortex lat-o tices for isotropic harmonic potentials using the radial basis function collocation method with predictor-corrector (PC) continuation method. Finally, we implement a two-parameter continuation algorithm to observe the dynamical formations of vor- tices under the harmonic potential, harmonic-plus-optical lattice potentials and the harmonic-plus-quartic potentials for various parameters. In the last part, we present an explicit local radial basis function collocation method (LRBFCM) to analyze the temperature distribution during laser heating therapy. Two examples are studied. In the first example we demonstrate the cor- rectness of the proposed method and in the second example, our numerical results show that the local radial basis function collocation method can efficiently solve the three-dimensional bioheat equation and it is easy to implement on complex domains.
在本論文中, 我們研究量身定做的有限點法和徑向基底函數配點法求解偏微分方程式,包括薛丁格方程式和生物傳熱方程式。 在第一部分中, 將不含時薛丁格方程式當作奇異擾動問題。 在位能為非負與齊次, 且次數大於零的假設下, 當奇異參數接近零時, 我們提出兩個定理來描述此問題在無界域上的行為以及兩個在有界域上的猜想。 我們還提出了兩種量身定做的有限點格式以求得此問題之數值解。 對於兩種特殊實例, 我們的數值解與理論分析一致。 此外, 我們運用第二種量身定做的有限點格式的非精確解法並且比較各式預處理元與迭代法的效能。 在第二部分, 我們研究在各種不同外加位能阱中, 旋轉玻色-愛因斯坦凝聚中的渦漩結構。 我們使用徑向基底函數配點法來求得描述玻色-愛因斯坦凝聚的耦合非線性方程之解。眾所周知, 徑向基底函數配點法的準確度和形狀參數有密切關連。 一開始, 我們給予解線性薛丁格方程式時選擇形狀參數的規則。 其次, 我們使用徑向基底函數配點法與預測-修正延續法在數值上模擬同向性簡諧位能阱下之渦漩晶格。最後, 我們實作雙參數延續演算法以觀察在簡諧位能阱, 與各種不同參數之簡諧加光晶格位能阱及簡諧加四次位能阱下旋轉玻色-愛因斯坦凝聚中渦漩的動態形成。 在最後的部分, 我們提出了顯性局部徑向基底函數配點法來分析雷射加熱治療時的溫度分佈。我們研究二個例子。 在第一例中, 我們證明了該方法的正確性而在第二例中, 我們的計算結果顯示, 局部徑向基底函數配點法可以有效求解三維生物傳熱方程式, 並且此方法在複雜域上的問題實作容易。
URI: http://hdl.handle.net/11455/90451
文章公開時間: 2017-07-16
Appears in Collections:應用數學系所

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